Question

Let $f\left(x\right)={\cos }^{2}x+{\cos }^2(x+\frac{\pi }{3})+{\cos }^2(x-\frac{\pi }{3})$ and $g\left(\frac{3}{2}\right)=1$ then the graph of $y=g\left(f\right(x\left)\right)$ is

• A. A circle
• B. A parabola
• C. A straight line
• D. None of these

Answer 1

Answer: (C)

A straight line

$f\left(x\right)={\cos }^2\left(x\right)+{\cos }^2(x+\pi /3)+{\cos }^2(x-\pi /3)=3/2$ (after simplification)
$\therefore g\left(x\right)=g\left(3/2\right)=1$ using $({\cos }^2\theta =\frac{1+\cos 2\theta }{2})$
$\Rightarrow y=g\left(f\right(x)=1$ which represents a straight line